A splitting theorem on smooth metric measure spaces

We consider a smooth metric measure space (M, g, e ^?f dv). Let ?? _f be its weighted Laplacian. Assuming that ??₁(?? _f ) is positive and the m-dimensional Bakry-émery curvature is bounded below in terms of ??₁(?? _f ), we prove a splitting theorem for (M, g, e ^?f dv). This theorem generalizes previous results by Lam and Li-Wang (Trans Am Math Soc 362:5043?C5062, 2010; J Diff Geom 58:501?C534, 2001; see also J Diff Geom 62:143?C162, 2002).     

On R-robustly entropy-expansive diffeomorphisms

For a homoclinic class H(p _f ) of f ?? Diff¹(M), f?OH(p _f ) is called R-robustly entropy-expansive if for g in a locally residual subset around f, the set ?? _? (x) = {y ?? M: dist(g ⁿ (x), g ⁿ (y)) ?? g3 (?n ?? ?)} has zero topological entropy for each x ?? H(p _g ). We prove that there exists an open and dense set around f such that for every g in it, H(p _g ) admits a dominated splitting of the form E ?? F ₁ ?? ... ?? F _k ?? G where all of F _i are one-dimensional and non-hyperbolic, which extends a result of Pacifico and Vieitez for robustly entropy-expansive diffeomorphisms. Some relevant consequences are also shown.     

Construction of nonseparable orthonormal compactly supported wavelet bases for L2（Rd）

Suppose M and N are two r × r and s × s dilation matrices,respectively.Let ΓM and ΓN represent the complete sets of representatives of distinct cosets of the quotient groups M-TZr/Zr and N-TZs/Zs,respe...     

Benz planes of Dembowski type and groups of tangential projectivities

Using the tangential relation we introduce in Benz planes M of Dembowski type, which generalize the Benz planes over algebras of characteristic 2, the group ?? of tangential perspectivities. We prove that these groups have the same behaviour as the classical groups of projectivities if any tangential perspectivity is induced by an automorphism of M. As permutation groups of a circle onto itself the groups ?? essentially differs from the classical groups of projectivities. If M is a Laguerre plane of Dembowski type, then ?? is always sharply 3-transitive. For Minkowski planes of Dembowski type ?? is at least 2-transitive. If M is a finite Benz plane of order 2 ^s , then ?? is isomorphic to the group PGL ₂(2 ^s ) in its sharply 3-transitive representation.     

Eigenvectors and cyclic vectors of bilateral weighted shifts. II: Simply invariant subspaces

Let T be an injective bilateral weighted shift onl ² thought as "multiplication by λ" on a space of formal Laurent series L²(β). (a) If L²(β) is contained in a space of quasi-analytic class of functions, then the point spectrum σ_p(T^?) of T^? contains a circle and the cyclic invariant subspaceM _f of T generated by f is simply invariant (i.e., ∩{(T^k M _f)^?: k ≥ 0}= {0}) for each f in L²(β); (b) If L²(β) contains a non-quasi-analytic class of functions (defined on a circle г) of a certain type related with the weight sequence of T, then there exists f in L²(ß) such thatM _f is a non-trivial doubly invariant subspace (i.e., (TM _f)^? =M _f); furthermore, if г ? σ_p(T^*), then σ_p (T^*) = г and f can be chosen so that σ_p([T∣M _f]^*) = г?{α}, for some α ε г. Several examples show that the gap between operators satisfying (a) and operators satisfying (b) is rather small.     

On the Behaviour of the spectral characteristic of Feigenbaum’s map

Let T _g : [?1, 1] ?? [?1, 1] be the Feigenbaum map. It is well known that T _g has a Cantor-type attractor F and a unique invariant measure ??₀ supported on F. The corresponding unitary operator (U _g ??)(x) = ??(g(x)) has pure point spectrum consisting of eigenvalues ?? _n,r , n ?? 1, 0 ?? r ?? 2 ^n?1 ? 1 with eigenfunctions e _r ⁽ⁿ⁾ (x). Suppose that f ?? C ¹([?1, 1]), f?? is absolutely continuous on [?1, 1] and f?? ?? L ^p ([?1, 1], d??₀), p > 1. Consider the sum of the amplitudes of the spectral measure of f: $$ Sn(f): = \sum\limits_{r = 0}^{2^n - 1} {|\rho _r^{(n)} |^2 ,\rho _r^{(n)} = \int\limits_{ - 1}^1 {f(x)\overline {e_r^{(n)} (x)} d\mu _o } } (x). $$ Using the thermodynamic formalism for T _g we prove that S _n (f) ?? 2^?n q ⁿ , as n ?? ??, where the constant q ?? (0, 1) does not depend on f.     

The barycenter property for robust and generic diffeomorphisms

Let f:M~d→M~d(d≥2) be a diffeomorphism on a compact C~∞ manifold on M.If a diffeomorphism f belongs to the C~1-interior of the set of all diffeomorphisms having the barycenter property,then f is Ω-stable.Moreover,if a generic diffeomorphism f has the barycenter property,then f is Ω-stable.We also apply our results to volume preserving diffeomorphisms.     

On bijections of Lorentz manifolds,which leave the class of spacelike paths invariant

Suppose that (M, g) and (M′, g′) are Lorentz manifolds, and that f: M → M′ is a bijection, such that f and f^-1 preserve spacelike paths (f: M → M′ has this property, if for any spacelike path γ: J → M in (M ,g), the composition fγ: J → M′ is a spacelike path in (M′, g′)). Then f is a (manifold-) homeomorphism.This statement is the ‘spacelike’ version of an analogous ‘timelike’ theorem (Hawking, King and McCarthy [6] and Göbel [2] for strongly causal, and Malament [10] for general Lorentz manifolds).With this result it is possible to prove a conjecture of Göbel [3] which states that every bijection between time-orientable n-dimensional (n ? 3) Lorentz manifolds which preserves spacelike paths is a conformal C^∞-diffeomorphism.     

Homotopy Sphere Representations for Matroids

For any rank r oriented matroid M, a construction is given of a ??topological representation?? of M by an arrangement of homotopy spheres in a simplicial complex which is homotopy equivalent to S ^r?C1 . The construction is completely explicit and depends only on a choice of maximal flag in M. If M is orientable, then all Folkman-Lawrence representations of all orientations of M embed in this representation in a homotopically nice way.     

On Local Constancy of Topological Entropy for Certain Partially Hyperbolic Diffeomorphisms

Let f: M → M be a partially hyperbolic diffeomorphism on a closed Riemannian manifold with uniformly compact center foliation. We show that if the center foliation of f is of dimension one then the topological entropy is constant on a small C¹ neighborhood of f.     

L 1-Approximation and Finding Solutions with Small Support

In this paper, we study an interesting property of L ¹-approximation. For many subspaces M, there exist ?? ^?(M)>0 with the following property: if f vanishes off a set of measure at most ?? ^?(M), then the zero function is a best L ¹-approximant to f from M. We explain this phenomenon, provide estimates for ?? ^?(M) in many cases, and present some open questions.     

Invertible linear maps preserving {1}-inverses of matrices over PID

Let R be a PID,chR = 2,n > 1, M_n(R) be then xn full matrix algebra over R.f denotes any invertible linear map preserving {1}-inverses from M_n(R) to itself. In this paper, we have proven thatf is an invertible linear map on M_n(R) preserving {1}-inverses if and only iff satisfies any one of the following two conditions: (i) there exists a matrixP ? GL _n(R) such thatf(A) =PAP ^?1 for allA ? M _n(R), (ii) there exists a matrixP ? GL _n(R) such thatf(A) =PA ^t P^?1 forA ? M _n(R).     

A note on shadowing with chain transitivity

Let f : X → X be a continuous map of a compact metric space X. The map f induces in a natural way a map f_M on the space M(X) of probability measures on X, and a transformation f_K on the space K(X) of closed subsets of X. In this paper, we show that if (X, f) is a chain transitive system with shadowing property, then exactly one of the following two statements holds:

(a)

fⁿ and (f_K)ⁿ are syndetically sensitive for all n ? 1.

(b)

fⁿ and (f_K)ⁿ are equicontinuous for all n ? 1.

In particular, we show that for a continuous map f : X → X of a compact metric space X with infinite elements, if f is a chain transitive map with the shadowing property, then fⁿ and (f_K)ⁿ are syndetically sensitive for all n ? 1. Also, we show that if f_M (resp. f_K) is chain transitive and syndetically sensitive, and f_M (resp. f_K) has the shadowing property, then f is sensitive.In addition, we introduce the notion of ergodical sensitivity and present a sufficient condition for a chain transitive system (X, f) (resp. (M(X), f_M)) to be ergodically sensitive. As an application, we show that for a L-hyperbolic homeomorphism f of a compact metric space X, if f has the AASP, then fⁿ is syndetically sensitive and multi-sensitive for all n ? 1.     

Positive Results and Counterexamples in Comonotone Approximation

We estimate the degree of comonotone polynomial approximation of continuous functions f, on [?1,1], that change monotonicity s??1 times in the interval, when the degree of unconstrained polynomial approximation E _n (f)??n ^??? , n??1. We ask whether the degree of comonotone approximation is necessarily ??c(??,s)n ^??? , n??1, and if not, what can be said. It turns out that for each s??1, there is an exceptional set A _s of ????s for which the above estimate cannot be achieved.     

Existence of a double random integral with respect to stable measures

In this paper, the authors study a double random integral of the form ∫₀¹∫₀¹f(s,t) M(ds) M(dt), where M(0,t) is a stable process with independent increments. Basically, the Wiener approach is used, and the existence of the above integral is established for a wide class of functions f.     

On the coefficients of a bounded univalent analytic function

LetF _M denote the class of univalent analytic functionsf in |z|<1 with the expansionf (z)=z+a ₂ z ²+a ₃ z ³+... and |f(z)|?M in |z|<1. In this note I derive a rough bound for alln-th coefficients and a more accurate bound for all the third coefficients of functionsf belonging toF _M.     

Atomic Decomposition of Hardy Type Spaces on Certain Noncompact Manifolds

In this paper we consider a complete connected noncompact Riemannian manifold M with bounded geometry and spectral gap. We prove that the Hardy type spaces X ^k (M), introduced in a previous paper of the authors, have an atomic characterization. An atom in X ^k (M) is an atom in the Hardy space H ¹(M) introduced by Carbonaro, Mauceri, and Meda, satisfying an ??infinite dimensional?? cancellation condition. As an application, we prove that the Riesz transforms of even order $\nabla^{2k} \mathcal{L}^{-k}$ map X ^k (M) into L ¹(T _2k M).     

One-basedness and groups of the form G/G 00

We initiate a geometric stability study of groups of the form G/G ⁰⁰, where G is a 1-dimensional definably compact, definably connected, definable group in a real closed field M. We consider an enriched structure M?? with a predicate for G ⁰⁰ and check 1-basedness or non-1-basedness for G/G ⁰⁰, where G is an additive truncation of M, a multiplicative truncation of M, SO ₂(M) or one of its truncations; such groups G/G ⁰⁰ are now interpretable in M??. We prove that the only 1-based groups are those where G is a sufficiently ??big?? multiplicative truncation, and we relate the results obtained to valuation theory. In the last section we extend our results to ind-hyperdefinable groups constructed from those above.     

Stabilizers and orbits of smooth functions

Let be a smooth function such that f(0)=0. We give a condition J(id) on f when for arbitrary preserving orientation diffeomorphism such that ?(0)=0 the function ?○f is right equivalent to f, i.e. there exists a diffeomorphism such that ?○f=f○h at 0∈Rm. The requirement is that f belongs to its Jacobi ideal. This property is rather general: it is invariant with respect to the stable equivalence of singularities, and holds for non-degenerated, simple, and many other singularities.We also globalize this result as follows. Let M be a smooth compact manifold, a surjective smooth function, DM the group of diffeomorphisms of M, and the group of diffeomorphisms of R that have compact support and leave [0,1] invariant. There are two natural right and left-right actions of DM and on C^∞(M,R). Let SM(f), SMR(f), OM(f), and OMR(f) be the corresponding stabilizers and orbits of f with respect to these actions. We prove that if f satisfies J(id) at each critical point and has additional mild properties, then the following homotopy equivalences hold: SM(f)≈SMR(f) and OM(f)≈OMR(f). Similar results are obtained for smooth mappings M→S¹.     

Direct iterative methods for least-squares solutions to singular operator equations

Least-squares consistency and convergence of iterative schemes are investigated for singular operator equations (1) Tx = f, where T is a bounded linear operator from a Banach space to a Hilbert space. A direct splitting of T into T = M ? N is then used to obtain the iterative formula (2) x^(k+1) = M^?Nx^(k) + M^?f, where M^? is a least-squares generalized inverse of M. Cone monotonicity is used to investigate convergence of (2) to a least-squares solutions to (1), extending results given for the matrix case given by Berman and Plemmons.